Decoding Logarithmic Expressions A Step-by-Step Solution

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Hey guys! Let's dive into the fascinating world of logarithms and break down this complex logarithmic expression step by step. We'll explore the fundamental concepts and properties of logarithms, making it super easy to understand. So, buckle up and get ready for a logarithmic adventure!

Understanding Logarithms

Before we tackle the main expression, let's make sure we're all on the same page with the basics of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simple terms, if we have an equation like b^x = y, the logarithm of y to the base b is x. This is written as log_b(y) = x.

The base of a logarithm is the number that is raised to a power. For example, in log_4(8), the base is 4. The argument of a logarithm is the number whose logarithm is being taken. In log_4(8), the argument is 8. Understanding these definitions is crucial for simplifying logarithmic expressions. For example, let’s consider log_2(8). This asks the question: “To what power must we raise 2 to get 8?” Since 2^3 = 8, the answer is 3. So, log_2(8) = 3. This foundational knowledge will be key as we tackle more complex parts of the expression. Remember, logarithms are just another way of expressing exponents, and with a solid grasp of this concept, we can simplify almost any logarithmic problem.

Key Properties of Logarithms

There are several key properties of logarithms that will help us simplify the expression. Let's take a look at a few:

  1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
  2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  3. Power Rule: log_b(m^p) = p * log_b(m)
  4. Change of Base Formula: log_b(a) = log_c(a) / log_c(b)

These properties allow us to manipulate logarithmic expressions and break them down into simpler terms. Mastering these properties is like having a superpower in the world of logarithms. For example, the product rule tells us that the logarithm of a product is the sum of the logarithms. This can be incredibly useful for breaking down complex arguments into simpler components. Similarly, the quotient rule helps us deal with division inside logarithms by turning them into differences. The power rule is particularly handy when the argument involves an exponent, allowing us to bring the exponent down as a multiplier. Lastly, the change of base formula is essential when dealing with logarithms of different bases, enabling us to convert them to a common base for easier calculation. As we go through the steps, you'll see these properties in action, and you'll get a better feel for how to use them.

Breaking Down the Expression

Now, let's get to the main event! We have the following expression:

Log_4 8 + Log_4 8 + Log_2 8 - Log 1000 + Log_{13} 169 Log_9 18 + Log_9 (1/2) - Log_5 125 + Log_3 1 + Log_4 256 Log_2 1024 + Log_9 1.9 - Log_6 216 + Log_7 49 - Log_3 81 Log_{15} 225 - Log_{14} 196 + Log_3 81.243 - Log_3 729

It looks intimidating, but don't worry! We're going to tackle it one piece at a time. The key here is to take it slow and apply the properties we just discussed. We'll break down each term, simplify it, and then put it all back together. Think of it like solving a puzzle – each logarithmic term is a piece, and our goal is to fit them together in the right way. We'll start with the easier ones and gradually move to the more complex ones. By doing this, we avoid getting overwhelmed and can focus on the specifics of each part. So, let's put on our detective hats and start decoding!

Step 1 Simplify Individual Logarithmic Terms

Let's start by simplifying each logarithmic term individually:

  • Log_4 8: We need to find the power to which we must raise 4 to get 8. Since 4^(3/2) = 8, Log_4 8 = 3/2
  • Log_2 8: We need to find the power to which we must raise 2 to get 8. Since 2^3 = 8, Log_2 8 = 3
  • Log 1000: This is a common logarithm (base 10). We need to find the power to which we must raise 10 to get 1000. Since 10^3 = 1000, Log 1000 = 3
  • Log_13} 169 We need to find the power to which we must raise 13 to get 169. Since 13^2 = 169, Log_{13 169 = 2
  • Log_9 18 + Log_9 (1/2): Using the product rule, we can combine these into a single logarithm: Log_9 (18 * 1/2) = Log_9 9 = 1
  • Log_5 125: We need to find the power to which we must raise 5 to get 125. Since 5^3 = 125, Log_5 125 = 3
  • Log_3 1: Any number raised to the power of 0 is 1, so Log_3 1 = 0
  • Log_4 256: We need to find the power to which we must raise 4 to get 256. Since 4^4 = 256, Log_4 256 = 4
  • Log_2 1024: We need to find the power to which we must raise 2 to get 1024. Since 2^10 = 1024, Log_2 1024 = 10
  • Log_9 1.9: This one is a bit tricky and doesn't simplify to a whole number. We'll keep it as Log_9 1.9 for now.
  • Log_6 216: We need to find the power to which we must raise 6 to get 216. Since 6^3 = 216, Log_6 216 = 3
  • Log_7 49: We need to find the power to which we must raise 7 to get 49. Since 7^2 = 49, Log_7 49 = 2
  • Log_3 81: We need to find the power to which we must raise 3 to get 81. Since 3^4 = 81, Log_3 81 = 4
  • Log_15} 225 We need to find the power to which we must raise 15 to get 225. Since 15^2 = 225, Log_{15 225 = 2
  • Log_14} 196 We need to find the power to which we must raise 14 to get 196. Since 14^2 = 196, Log_{14 196 = 2
  • Log_3 81.243: This is similar to Log_9 1.9 and doesn't simplify to a whole number easily. We'll keep it as Log_3 81.243 for now.
  • Log_3 729: We need to find the power to which we must raise 3 to get 729. Since 3^6 = 729, Log_3 729 = 6

By going through each term individually, we've managed to simplify the expression significantly. This step-by-step approach is a powerful technique for tackling complex problems. Notice how we used the definition of a logarithm to convert each logarithmic term into a simpler numerical value. This process highlights the core idea that logarithms are just exponents in disguise. The key is to ask yourself,